Dynamic current amplification

Another way of specifying current amplification is as the ratio of the difference in I_C to the difference in I_B. Abbreviate the words “the difference in” by the letter d. Then, according to this second definition:
Current amplification = dIC/dIB
A graph of collector current versus base current (I_C vs I_B) for a hypothetical transistor is shown in Fig. 1. Except that current, rather than voltage, is on the horizontal scale. Three different points are shown, corresponding to various bias values.

Fig. 1: Three different transistor bias points. See text for discussion.

The ratio dIC/dIB is different for each of the points in this graph. Geometrically, dI_C/dI_B at a given point is the slope of a line tangent to the curve at that point. The tangent line for point B in Fig. 1 is a dotted, straight line; the tangent lines for points A and C lie right along the curve. The steeper the slope of the line, the greater is dI_C/dI_B. Point A provides the highest dI_C/dI_B , as long as the input signal is small. This value is very close to H_FE. For small-signal amplification, point A represents a good bias level. Engineers would say that it’s a good operating point.
At point B, dI_C/dI_B is smaller than at point A. (It might actually be less than 1.) At point C, dI_C/dI_B is practically zero. Transistors are rarely biased at these points.